Mean-field model of interacting quasilocalized excitations in glasses

Structural glasses feature quasilocalized excitations whose frequencies $\omega$ follow a universal density of states ${\cal D}(\omega)\!\sim\!\omega^4$. Yet, the underlying physics behind this universality is not yet fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles described collectively as anharmonic oscillators, which are coupled among themselves through random interactions that are characterized by a disorder strength $J$. In addition, the oscillators interact with a uniform constant external field $h$. We numerically show that the model generically gives rise to a gapless ${\cal D}(\omega)\!=\!A_{\rm g}\,\omega^4$ density of states, where the prefactor $A_{\rm g}(h,J)$ depends on the state of disorder. It is further shown that $A_{\rm g}(h,J)$ is a nonmonotonic function of $J$ for a fixed $h\!>\!0$, varying exponentially with $-1/J^2$ in the small $J$ regime and decays as a power-law in the large $J$ regime. We discuss the possible relations between our findings and available observations in structural glasses, along with delineating some future research directions.

Nearly two decades ago, Gurevich, Parshin and Schober (GPS) put forward a three-dimensional (d = 3, whered is the spatial dimension) lattice model [19], aimed at resolving the vibrational density of states of QLMs. The model assumes QLMs to exist and to be described as anharmonic oscillators -meant to represent small, spatially-localized sets of particles -that populate lattice sites, and interact with each other via random couplings. The latter are characterized by strength J that follows the same ∼ r −d spatial decay of linearelastic dipole-dipole interactions, where r is the distance between the oscillators. GPS showed numerically that the model's vibrational spectrum indeed grows from zero frequency as D(ω) ∼ ω 4 [19], and these numerical results have been rationalized using a phenomenological theory [19][20][21].
rently lack insight into the origin of QLMs' statisticalmechanical properties. Moreover, recent progress in studying computer glass-formers revealed intriguing properties of QLMs, e.g. the dependence of the amplitude A g of the ω 4 universal law on the state of glassy disorder [17,18,27], which are not yet fully understood. In this work we study numerically the spectral properties of the mean-field variant of GPS's lattice model, obtained by taking the limit of infinite spatial dimension (d → ∞), and subjecting the model's oscillators to a constant external field h. A similar mean-field model was studied first by Kühn and Horstmann [28] in the context of lowtemperature glassy anomalies [29][30][31]. We therefore refer to our model hereafter as the KHGPS model. In this work, we show that the low-frequency spec-arXiv:2010.11180v2 [cond-mat.dis-nn] 29 Oct 2020 trum of the KHGPS model [32] -to be explicitly formulated below -generically follows D(ω) = A g ω 4 , as is widely observed in particle-based computer glassformers, cf. Fig. 1b. We further show that the nonuniversal prefactor A g (h, J), which captures the characteristic frequency and number density of soft QLMs in particle-based computer glass-formers [17,33], reveals rich dependence on the model parameters h and J. In particular, A g (h, J) features two scaling regimes for a fixed field h > 0; in the small J regime, log A g (h, J) ∼ −1/J 2 , reminiscent of its behavior in computer glass models as a function of the temeprature T p at which the glass falls out of equilibrium [17,27], cf. inset of Fig. 1b. At larger coupling strength, we find is found to be a nonmonotonic function of the strength of random couplings J for h > 0. We discuss these results in light of recent observations in computer glass-forming models and delineate some future research directions.
Model.-QLMs in glasses have been shown to feature large displacements inside a localized core of a few atomic distances in linear size, accompanied by power-law decaying dipolar displacements away from the core. QLMs also feature low vibrational frequencies, i.e. they represent particularly soft regions inside a glass, and are randomly distributed in space. The main question we aim at addressing in this work is whether one can develop a relatively simple mean-field model, using this physical picture of QLMs as an input, to obtain their universal density of states D(ω) = A g ω 4 and to gain insight into the properties of the non-universal prefactor A g .
To this aim, closely following GPS, we adopt a coarsegrained picture in which QLMs are anharmonic oscillators embedded inside an elastic medium. The elastic medium mediates interactions between the QLMs and can also affect them directly. We consider N anharmonic oscillators, each described by a generalized coordinate x i , whose Hamiltonian takes the form The oscillators in Eq. (1) are characterized by random harmonic stiffnesses κ i , which follow a distribution p(κ) that reflects their softness (see details below). They also feature fourth order stabilizing anharmonicities, whose strength A is set to unity hereafter. Each anharmonic In light of the KHGPS Hamiltonian in Eq. (1), our first goal is to understand whether, and if so under what conditions, rather featureless distributions p(κ) lead to D(ω) ∼ ω 4 , where ω 2 are the stiffnesses that characterize the minima of H. While formally we are only interested in the zero temperature (T → 0) properties of the model, i.e. in the statistical properties of the minima of H, one can hypothesize that p(κ) represents in some sense a high T state of a glass-forming liquid, and that the H minimization procedure corresponds in some sense to the self-organization processes a glass undergoes during a quench to a low T state. The only constraint we impose on p(κ) is that it does not feature a gap around κ = 0, i.e. that there exists no κ g > 0 for which p(κ) vanishes over 0 < κ < κ g . This constraint highlights the fact that our model does not aim at explaining the existence of arbitrarily soft excitations in glasses, but rather their statistical properties.
In pragmatic terms, we initialize N = 16000 oscillators placed at x i = 0, and assign values for the parameters pair (h, J). We next draw the stiffnesses κ i from a uniform distribution over the interval [0,1], and draw the couplings J ij from a Gaussian distribution with width J/ √ N . We then initialize the Hamiltonian given in Eq. (1), apply a small, random perturbation of the oscillators coordinates, of average magnitude of 10 −2 , and follow it by a standard nonlinear conjugate gradient minimization of the Hamiltonian. The Hessian of the Hamiltonian, M ij ≡ ∂ 2 H/∂x i ∂x j , is evaluated and diagonalized upon reaching a minimum. This procedure is repeated 1150 times for each pair (h, J), and the statistics of the respective spectra are analyzed.
Results.-We consider first the vibrational spectra of the KHGPS model in the absence of external field, i.e. for h = 0. In Fig. 2a we plot D(ω/ √ J) vs. the dimensionless frequency ω/ √ J. We find that D(ω) ∼ ω 4 (with measurable finite-size effects for smaller J, as also known to emerge in small particulate computer glasses [34]), for all couplings J employed (indicated in the figure legend). The existence of characteristic frequency ∼ √ J is made apparent by the low-frequency collapse of D(ω) for different coupling strengths J. Consequently, we predict that A g ∼ J −5/2 , and expect it to remain valid for large enough J, even for h > 0, as will be demonstrated below.
We next subject the oscillators to an external field h = 0.01, and plot D(ω) in Fig. 2b for 3 values of J (indicated in the legend). As opposed to the zero-field case discussed above, we see that turning on the external field renders the prefactor A g (h, J) nonmonotonic with J, with a maximum residing between J = 0.05 and J = 0.2. To better resolve the J-dependence of the prefactor A g (h, J), we simulate the KHGPS model under an external field h = 0.01 and several values of J ∈ [0.035, 1.6]. To each averaged vibrational spectrum, we estimate A g (h, J) by fitting the ω 4 tails for all couplings J. The results are shown in Fig. 3. We find that A g assumes a maximum around J max ≈ 0.1, for h = 0.01.
Interestingly, for J < J max we find that log A g ∼ −1/J 2 , as indicated by the continuous line in Fig. 3a. This behavior is reminiscent of the dependence of A g on the parent temperature T p as seen in particulate computer glasses, cf. inset of Fig. 1b. Finally, our aforementioned expectation A g ∼ J −5/2 for large J is validated in Fig. 3b. To summarize, the prefactor A g (h, J) of the KHGPS model's ∼ ω 4 vibrational spectrum, for h > 0, appears to follow The precise dependence of J max on h is left for future investigations.
Discussion.-In this work, we numerically studied a mean-field model of interacting quasilocalized excitations, termed the KHGPS model and defined by the Hamiltonian in Eq. (1). A major result of our analysis is that local minima of the model [32] appear to robustly feature D(ω) ∼ ω 4 vibrational spectra, independently of the input parameters h and J, as long as p(κ) is gapless. As such, the KHGPS Hamiltonian in Eq.
In light of these previous efforts, our results appear to support -and further highlight -GPS's suggestion [19,20] that stabilizing anharmonicities -absent from the aforementioned mean-field models -constitute a necessary physical ingredient for observing the universal ∼ ω 4 law in this class of mean-field models. The universality of the D(ω) ∼ ω 4 law implies that all non-universal material properties crucially depend on the prefactor A g in D(ω) = A g ω 4 , as discussed extensively in recent literature [17,18,39,40]. Therefore, we set out to study the dependence of A g on the model's parameters h and J.
We found that A g (h, J) exhibits two distinct scaling regimes that depend on the value of the constant field h. At small J values, A g (h, J) is found to vary exponentially with −1/J 2 , which is reminiscent of the variation of A g with the temperature T p at which a glass falls out of equilibrium, see inset of Fig. 1b. The two observations may indeed be related if the model's parameter J can be somehow mapped onto T p ; such a mapping may be physically sensible as both J and T p affect the strength of disorder in a glass.
Our results further support that A g (h, J) varies exponentially with −f (h)/J 2 , where the function f (h) satisfies f (h = 0) = 0, as indicated by the absence of an exponential regime for h = 0. We also found that for large J values, where 'large' should be defined relative to J max (h), a different scaling behavior in the form of A g (h, J) ∼ J −5/2 emerges. In this scaling regime, the model is characterized by a single frequency scale ∼ √ J; the A g (h, J) ∼ J −5/2 relation immediately follows from the existence of this single scale, in conjunction with D(ω) = A g ω 4 . Consequently, the non-universal prefactor A g (h, J) exhibits a nonmonotonic behavior, with an h-dependent crossover between two different scaling regimes, cf. Fig. 3.
Our findings give rise to several interesting research directions. First, it would be most useful to obtain an analytic solution of the model, predicting the numerical observations. Such a solution may provide deeper understanding of the statistical-mechanical properties of the KHGPS model, and better place it in the context of other mean-field models in statistical physics. Moreover, such a solution may shed light on the finite T behavior of the KHGPS model, not discussed here at all. We believe that obtaining this analytic solution is possible, and hope to report on it in the near future. Second, it would be most useful to further explore the analogy between the KHGPS model and finite-dimensional glasses, in particular better understanding the hypothesized relation between the minimization of the Hamiltonian and the self-organization processes taking place while a glass is quenched from a melt. Establishing such relations may clarify what mean-field models such as the KHGPS one can teach us about the physics of glasses.